Yes, Starship needs legs to land on the Moon or Mars. They will be lighter than legs for Earth landings, as Moon/Mars have lower gravity.
Not sure about the amount of fuel needed to land, as there is much less atmosphere on Mars, and none on the Moon. I would still guess that it needs less fuel there as well.
An atmosphere reduces the amount of fuel needed to land, because you can use aerobraking to slow down rather than carry fuel to do it. See the Apollo return capsule, which landed without any rockets, only parachutes and a heat shield.
Yes, also the lunar variant won't need a heat shield so that will compensate for the extra leg mass. Not sure what their plans are for a heat shield for Mars.
They're planning on catching starship as well however, as the in-orbit refueling will require a lot of starship launches that aren't going anywhere except orbit and back.
But the other Starships don't and adding them would make a big difference.
Since you need a huge fuel tank in orbit that needs to be refilled by multiple starships to refuel Starship HLS to actually land on the moon adding legs on those refueling Starships would decrease payload capacity, and thus you'd need even more Starships to refuel the tank for a single lunar landing.
